Financial advisory system

ABSTRACT

A financial advisory system is provided. According to one aspect of the present invention, return scenarios for optimized portfolio allocations are simulated interactively to facilitate financial product selection. Return scenarios for each asset class of a plurality of asset classes are generated based upon estimated future scenarios of one or more economic factors. A mapping from each financial product of an available set of financial products onto one or more asset classes of the plurality of asset classes is created by determining exposures of the available set of financial products to each asset class of the plurality of asset classes. In this way, the expected returns and correlations of a plurality of financial products are generated and used to produce optimized portfolios of financial products. Return scenarios are simulated for one or more portfolios including combinations of financial products from the available set of financial products based upon the mapping.

This is a Divisional of U.S. patent application Ser. No. 09/495,982,filed on Feb. 1, 2000, currently pending, which is aContinuation-In-Part of application Ser. No. 08/982,942, filed on Dec.2, 1997, now issued as U.S. Pat. No. 6,021,397.

COPYRIGHT NOTICE

Contained herein is material that is subject to copyright protection.The copyright owner has no objection to the facsimile reproduction ofthe patent disclosure by any person as it appears in the Patent andTrademark Office patent files or records, but otherwise reserves allrights to the copyright whatsoever.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The invention relates generally to the field of financial advisoryservices. More particularly, the invention relates to a system foradvising a user regarding feasible and optimal portfolio allocationsamong a set of available financial products.

2. Description of the Related Art

During the 1980's, a significant trend emerged in retirement savings.Traditional defined benefit plan assets began shifting towardsemployee-directed defined contribution plans like 401(k). As this trendcontinues, many individual investors will ultimately become responsiblefor managing their own retirement investments. However, many people arenot well-equipped to make informed investment decisions. Further, thenumber and diversity of investment options available to individuals israpidly increasing, thereby making investment decisions more complex bythe day.

Many investment software packages claim to help individuals plan for asecure retirement, or some other intermediate goal. However, typicalprior art investment software packages are limited in several ways. Forexample, some packages provide generic asset-allocation suggestions(typically in the form of a pie-chart) and leave the investor to findthe actual combination of financial products that meets the suggestedasset allocation. However, many investments available to individualinvestors, such as mutual funds, cannot easily be categorized into anyone generic asset class category. Rather, mutual funds are typically amix of many different asset classes. This property of mutual fundscomplicates the selection of appropriate instruments to realize adesired asset allocation.

Further, some prior art programs, typically referred to as “retirementcalculators,” require the user to provide estimates of future inflation,interest rates and the expected return on their investments. In thistype of prior art system, the user is likely, and is in fact encouraged,to simply increase the expected investment returns until their desiredportfolio value is achieved. As should be appreciated, one of theproblems with this type of program is that the user is likely to createan unattainable portfolio based on an unrealistic set of future economicscenarios. That is, the portfolio of financial products required toachieve the X % growth per year in order to meet the user's retirementgoal may not be available to the user. Further, the idealistic futureeconomic conditions assumed by the user, for example, 0% inflation and20% interest rates, may not be macroeconomically consistent. Typicalprior art investment packages simply allow the user to manipulateeconomic conditions until a desired result is achieved rather thanencouraging the user to focus on his/her own decisions regardinginvestment risk, savings rate, and retirement age within the context ofrealistic economic assumptions. Consequently, the so called “advice”rendered by many of the prior art investment software packages can bemisleading and impossible to implement in practice.

In addition, investment advice software in the prior art have variousother disadvantages which are overcome by the present invention.Notably, prior art systems typically do not provide realistic estimatesof the investment or retirement horizon risk-return tradeoff given auser's specific investments and financial circumstances. This makesinformed judgments about the appropriate level of investment risk verydifficult. Obtaining the appropriate level of investment risk (andreturn) is critical to the success of a long-term investment plan.

In view of the foregoing, what is needed is a financial advisory systemthat employs advanced financial techniques to provide financial adviceto individuals on how to reach specific financial goals. Morespecifically, it is desirable to provide a system that automaticallygenerates future-looking realistic economic and investment returnscenarios and allows a user to arrive at a feasible portfolio that meetsboth intermediate and long-term financial goals by a process ofoutcome-based risk profiling. In this manner, the user can focus onhis/her own decisions regarding investment risk, savings, and retirementage while interactively observing the impact of those decisions on therange of possible investment outcomes. Further, it is desirable that thefinancial advisory system create a feasible optimal portfolio thatmaximizes the utility function of the user by selecting financialproducts that are available to the user and that provides the highestpossible utility given the user's risk tolerance, investment horizon andsavings level. By utility what is meant is a function that determinesthe relative preferences of an individual for different combinations offinancial products based on one or more characteristics of the products(e.g., expected return, variance, etc.), and optionally one or moreparameters specific to the individual. Moreover, it is advantageous toperform plan monitoring on an ongoing basis to alert the user if thelikelihood of meeting their financial goals falls below a thresholdvalue or if their portfolio risk level becomes inconsistent with theirrisk preferences. Finally, it is desirable to provide specific advice tothe user regarding steps they can take to improve their chances ofmeeting their financial goals while taking into consideration the user'spersonal tradeoffs among risk, savings, and retirement age.

BRIEF SUMMARY OF THE INVENTION

A financial advisory system is described. According to one aspect of thepresent invention, return scenarios for optimized portfolio allocationsare simulated interactively to facilitate financial product selection.Return scenarios for each asset class of a plurality of asset classesare generated based upon estimated future scenarios of one or moreeconomic factors. A mapping from each financial product of an availableset of financial products onto one or more asset classes of theplurality of asset classes is created by determining exposures of theavailable set of financial products to each asset class of the pluralityof asset classes. In this way, the expected returns and correlations ofa plurality of financial products are generated and used to produceoptimized portfolios of financial products. Return scenarios aresimulated for one or more portfolios including combinations of financialproducts from the available set of financial products based upon themapping.

Other features of the present invention will be apparent from theaccompanying drawings and from the detailed description which follows.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

The present invention is illustrated by way of example, and not by wayof limitation, in the figures of the accompanying drawings and in whichlike reference numerals refer to similar elements and in which:

FIG. 1 illustrates a financial advisory system according to oneembodiment of the present invention.

FIG. 2 is an example of a typical computer system upon which oneembodiment of the present invention can be implemented.

FIG. 3 is a block diagram illustrating various analytic modulesaccording to one embodiment of the present invention.

FIG. 4 is a flow diagram illustrating core asset class scenariogeneration according to one embodiment of the present invention.

FIG. 5 is a flow diagram illustrating factor asset class scenariogeneration according to one embodiment of the present invention.

FIG. 6 is a flow diagram illustrating financial product exposuredetermination according to one embodiment of the present invention.

FIG. 7 is a flow diagram illustrating portfolio optimization accordingto one embodiment of the present invention.

FIG. 8 is a flow diagram illustrating plan monitoring processingaccording to one embodiment of the present invention.

DETAILED DESCRIPTION OF THE INVENTION

A financial advisory system is described. In embodiments of the presentinvention, a factor model approach is laid on top of a pricing kernelmodel to simulate returns of a plurality of asset classes, andultimately financial products, such as securities or portfolios ofsecurities. The term “financial products” as used herein refers to alegal representation of the right (often denoted as a claim or security)to provide or receive prospective future benefits under certain statedconditions. In any event, the forecasts may then be used for purposes ofproviding financial advisory services to a user. For example, suchforecasts are useful for selecting the composition of an optimizedportfolio (based on a utility function) from a set of availablefinancial products conditional on decisions and constraints provided bythe user.

Briefly, fundamental economic and financial forces are modeled using apricing kernel model that provides projected returns on a plurality ofasset classes (core asset classes) conditional on a set of statevariables that capture economic conditions. The core asset classes incombination with additional asset class estimates that are conditionedon the core asset classes comprise a model (hereinafter “the factormodel”) of a comprehensive set of asset classes that span the universeof typical investment products. A factor model is a return-generatingfunction that attributes the return on a financial product, such as asecurity, to the financial product's sensitivity to the movements ofvarious common economic factors. The factor model enables the system toassess how financial products and portfolios will respond to changes infactors or indices to which financial products are exposed. Theselection of asset classes may be tailored to address a narrow or broadrange of investors. For example, asset classes may be chosen that arerelevant only to a particular industry or asset classes may be chosen tospan the market range of a broad set of possible investments (e.g. allavailable mutual funds or individual equities). According to embodimentsof the present invention discussed herein, to reach the broadest segmentof individual investors, the asset classes selected as factors for thefactor model have been chosen to span the range of investments typicallyavailable to individual investors in mainstream mutual funds and definedcontribution plans.

After generating future scenarios for the factor model, financialproducts available to an investor may be mapped onto the factor model.To assure that a portfolio recommended by the system is attainable, itis preferable to generate investment scenarios that include only thosefinancial products that are available to the investor. The availablefinancial products may include, for example, a specific set of mutualfunds offered by an employer sponsored 401(k) program. In any event,this mapping of financial products onto the factor model is accomplishedby decomposing the returns of individual financial products intoexposures to the asset classes employed by the factor model. In thismanner, the system learns how each of the financial products availableto the user behave relative to the asset classes employed by the factormodel. In so doing, the system implicitly determines the constraints onfeasible exposures to different asset classes faced by an investor givena selected subset of financial products. Given this relationship betweenthe user's available financial products and the factor model, the systemmay generate feasible forward-looking investment scenarios. The systemmay further advise the user regarding actions that may be taken (e.g.,save more money, retire later, take on additional investment risk, seekopportunities to expand the investment set) to achieve certain financialgoals, such as particular retirement standard of living, accumulating adown payment for the purchase of a house, or saving enough money to senda child to college.

In the following description, for the purposes of explanation, numerousspecific details are set forth in order to provide a thoroughunderstanding of the present invention. It will be apparent, however, toone skilled in the art that the present invention may be practicedwithout some of these specific details. In other instances, well-knownstructures and devices are shown in block diagram form.

The present invention includes various steps, which will be describedbelow. The steps of the present invention may be embodied inmachine-executable instructions. The instructions can be used to cause ageneral-purpose or special-purpose processor that is programmed with theinstructions to perform the steps of the present invention.Alternatively, the steps of the present invention may be performed byspecific hardware components that contain hardwired logic for performingthe steps, or by any combination of programmed computer components andcustom hardware components.

The present invention may be provided as a computer program productwhich may include a machine-readable medium having stored thereoninstructions which may be used to program a computer (or otherelectronic devices) to perform a process according to the presentinvention. The machine-readable medium may include, but is not limitedto, floppy diskettes, optical disks, CD-ROMs, and magneto-optical disks,ROMs, RAMs, EPROMs, EEPROMs, magnet or optical cards, flash memory, orother type of media/machine-readable medium suitable for storingelectronic instructions. Moreover, the present invention may also bedownloaded as a computer program product, wherein the program may betransferred from a remote computer to a requesting computer by way ofdata signals embodied in a carrier wave or other propagation medium viaa communication link (e.g., a modem or network connection).

While, embodiments of the present invention will be described withreference to an financial advisory system, the method and apparatusdescribed herein are equally applicable to other types of assetallocation applications, financial planning applications, investmentadvisory services, financial product selection services, automatedfinancial product screening tools, such as electronic personal shoppingagents and the like.

System Overview

The present invention may be included within a client-server transactionbased financial advisory system 100 such as that illustrated in FIG. 1.According to the embodiment depicted in FIG. 1, the financial advisorysystem 100 includes a financial staging server 120, a broadcast server115, a content server 117, an AdviceServer™ 110 (AdviceServer™ is atrademark of Financial Engines, Inc., the assignee of the presentinvention), and a client 105.

The financial staging server 120 may serve as a primary staging andvalidation area for the publication of financial content. In thismanner, the financial staging server 120 acts as a data warehouse. Rawsource data, typically time series data, may be refined and processedinto analytically useful data on the financial staging server 120. On amonthly basis, or whatever the batch processing interval may be, thefinancial staging server 120 converts raw time series data obtained fromdata vendors from the specific vendor's format into a standard formatthat can be used throughout the financial advisory system 100. Variousfinancial engines may be run to generate data for validation and qualityassurance of the data received from the vendors. Additional engines maybe run to generate module inputs, model parameters, and intermediatecalculations needed by the system based on raw data received by thevendors. Any calibrations of the analytic data needed by the financialengines may be performed prior to publishing the final analytic data tothe broadcast server 115.

The broadcast server 115 is a database server. As such, it runs aninstance of a Relational Database Management System (RDBMS), such asMicrosoft SQL-Server™, Oracle™ or the like. The broadcast server 115provides a single point of access to all fund information and analyticdata. When advice servers such as AdviceServer 110 need data, they mayquery information from the broadcast server database. The broadcastserver 115 may also populate content servers, such as content server117, so remote implementations of the AdviceServer 110 need notcommunicate directly with the broadcast server 115.

The AdviceServer 110 is the primary provider of services for the client105. The AdviceServer 110 also acts as a proxy between external systems,such as external system 125, and the broadcast server 115 or the contentserver 117. The AdviceServer 110 is the central database repository forholding user profile and portfolio data. In this manner, ongoingportfolio analysis may be performed and alerts may be triggered, asdescribed further below.

According to the embodiment depicted, the user may interact with andreceive feedback from the financial advisory system 100 using clientsoftware which may be running within a browser application or as astandalone desktop application on the user's personal computer 105. Theclient software communicates with the AdviceServer 110 which acts as aHTTP server.

Overview of Exemplary User Interaction with the System

During an initial session with the financial advisory system 100,according to one embodiment of the present invention, the user mayprovide information regarding risk preferences, savings preferences,current age, gender, income, expected income growth, current accountbalances, current financial product holdings, current savings rate,retirement age goal, retirement income goals, available financialproducts, intermediate and long-term goals, constraints on fundholdings, liabilities, expected contributions, state and federal taxbracket (marginal and average). The user may provide information forthemselves and each profiled person in their household. This informationmay be saved in one or more files in the financial advisory system 100,preferably on one of the servers to allow ongoing plan monitoring to beperformed. In other embodiments of the present invention additionalinformation may be provided by the user, for example, estimates offuture social security benefits or anticipated inheritances.

Based on the user's current holdings the system may forecast aretirement income and graphically depict the current portfolio'sprojected growth and range of possible values over time.

The system may also provide the user with statistics regarding thelikelihood that they will be able to retire when they would like, giventhe projected returns on the user's current portfolio based upon thedata input by the user, including the user's current savings rate,retirement age goal, and investment holdings.

Based on models and calculations that will be discussed in more detailbelow, the financial advisory system 100 may provide an initialdiagnosis based upon the user's risk preference, savings rate, anddesired risk-return tradeoffs. This diagnosis can result in a series ofsuggested actions including: (1) rebalance the portfolio, (2) increasesavings, (3) retire later, or (4) adjust investment risk. An iterativeprocess may then begin in which the user may adjust his/her investmentrisk, savings rate, and/or retirement age and have the financialadvisory system 100 evaluate the projected performance of an optimizedportfolio given the financial products available to the user based onthe currently selected risk tolerance, investment horizon and savingsrate decisions. This process of the financial advisory system 100providing advice and/or feedback and the user adjusting risk, savings,and retirement age parameters may continue until the user has achieved adesired portfolio forecast and performance distribution. At this time,the user may chose to implement the optimal portfolio. The parametersand portfolio allocation may then be saved by the financial advisorysystem 100 for future user sessions.

As described further below, on an ongoing basis the financial advisorysystem 100 may evaluate the user's portfolio against one or morefinancial goals and may notify the user if progress towards any of thegoals has changed in a material way.

In subsequent user sessions with the financial advisory system 100, theuser's data (e.g., the user's profile information, account holdings,plan parameters, and tax information) may be retrieved from memory onthe AdviceServer 110, for example, and the current forecast for the oneor more goals may be presented to the user. Additionally, if the ongoingplan monitoring has generated any alerts, they may be presented to theuser at this time. Alternatively, alerts may be generated proactively bythe system and transmitted to the user via a telephone, email, fax, orstandard mail messaging system. Based upon the alerts generated by theongoing plan monitoring, the user may again begin the iterative processof adjusting the decision variables described above (e.g., risk level,savings rate, and retirement age) until the user is satisfied with thelikelihood of meeting his/her goal(s). To assure accurate portfoliotracking, if the personal data changes, the user may simply modify thedata upon which the financial advisory system's assumptions are based.For example, if the user's salary increases, this information should beupdated in the user's profile. Additionally, if the user's employer addsa new mutual fund to the company's 401(k) program, then the user shouldupdate the list of available financial products in the user profileinformation. This is important because the optimal allocation among theuser's available financial products may be impacted by the addition of anew mutual fund, for example. In one embodiment of the presentinvention, the financial advisory system 100 may be connected toexternal record-keeping systems at the user's employer that can provideautomatic updates to selected user information.

Advantageously, the user is never asked to predict the future withregard to interest rates, inflation, expected portfolio returns, orother difficult to estimate economic variables and parameters.Additionally, the optimal portfolio generated by the financial advisorysystem 100 is guaranteed to be attainable. That is, the optimalportfolio has been determined based upon the specific financial productsthat are available to the user.

An Exemplary Computer System

Having briefly described one embodiment of the financial advisory system100 and exemplary user interactions, a computer system 200 representingan exemplary client 105 or server in which features of the presentinvention may be implemented will now be described with reference toFIG. 2. Computer system 200 comprises a bus or other communication means201 for communicating information, and a processing means such asprocessor 202 coupled with bus 201 for processing information. Computersystem 200 further comprises a random access memory (RAM) or otherdynamic storage device 204 (referred to as main memory), coupled to bus201 for storing information and instructions to be executed by processor202. Main memory 204 also may be used for storing temporary variables orother intermediate information during execution of instructions byprocessor 202. Computer system 200 also comprises a read only memory(ROM) and/or other static storage device 206 coupled to bus 201 forstoring static information and instructions for processor 202.

A data storage device 207 such as a magnetic disk or optical disc andits corresponding drive may also be coupled to computer system 200 forstoring information and instructions. Computer system 200 can also becoupled via bus 201 to a display device 221, such as a cathode ray tube(CRT) or Liquid Crystal Display (LCD), for displaying information to acomputer user. For example, graphical depictions of expected portfolioperformance, asset allocation for an optimal portfolio, chartsindicating retirement age probabilities, and other data types may bepresented to the user on the display device 221. Typically, analphanumeric input device 222, including alphanumeric and other keys,may coupled to bus 201 for communicating information and/or commandselections to processor 202. Another type of user input device is cursorcontrol 223, such as a mouse, a trackball, or cursor direction keys forcommunicating direction information and command selections to processor202 and for controlling cursor movement on display 221.

A communication device 225 is also coupled to bus 201 for accessingremote servers, such as the AdviceServer 110, or other servers via theInternet, for example. The communication device 225 may include a modem,a network interface card, or other well known interface devices, such asthose used for coupling to an Ethernet, token ring, or other types ofnetworks. In any event, in this manner, the computer system 200 may becoupled to a number of clients and/or servers via a conventional networkinfrastructure, such as a company's Intranet and/or the Internet, forexample.

Exemplary Analytic Modules

FIG. 3 is a simplified block diagram illustrating exemplary analyticmodules of the financial advisory system 100 according to one embodimentof the present invention. According to the embodiment depicted, thefollowing modules are provided: a pricing module 305, a factor module310, a financial product mapping module 315, a tax adjustment module320, an annuitization module 325, a simulation processing module 330, aportfolio optimization module 340, a user interface (UI) module 345, anda plan monitoring module 350. It should be appreciated that thefunctionality described herein may be implemented in more or lessmodules than discussed below. Additionally, the modules andfunctionality may be distributed in various configurations among aclient system, such as client 105 and one or more server systems, suchas the financial staging server 120, the broadcast server 115, or theAdviceServer 110. The functionality of each of the exemplary moduleswill now be briefly described.

An “econometric model” is a statistical model that provides a means offorecasting the levels of certain variables referred to as “endogenousvariables,” conditional on the levels of certain other variables, knownas “exogenous variables,” and in some cases previously determined valuesof the endogenous variables (sometimes referred to as lagged dependentvariables). The pricing module 305 is an equilibrium econometric modelfor forecasting prices and returns (also referred to herein as “coreasset scenarios”) for a set of core asset classes. The pricing moduleprovides estimates of current levels and forecasts of economic factors(also known as state variables), upon which the estimates of core assetclass returns are based. According to one embodiment of the presentinvention, the economic factors may be represented with three exogenousstate variables, price inflation, a real short-term interest rate, anddividend growth. The three exogenous state variables may be fitted withautoregressive time series models to match historical moments of thecorresponding observed economic variables, as described further below.

In any event, the resulting core asset classes are the foundation forportfolio simulation and are designed to provide a coherent andinternally consistent (e.g., no arbitrage) set of returns. By arbitragewhat is meant is an opportunity to create a profitable tradingopportunity that involves no net investment and positive values in allstates of the world.

According to one embodiment, the core asset classes include short-termUS government bonds, long-term US government bonds, and US equities. Toexpand the core asset classes to cover the full range of possibleinvestments that people generally have access to, additional assetclasses may be incorporated into the pricing module 305 or theadditional asset classes may be included in the factor model 310 and beconditioned on the core asset classes, as discussed further below.

Based upon the core asset scenarios generated by the pricing module 305,the factor module 310 produces return scenarios (also referred to hereinas “factor model asset scenarios”) for a set of factor asset classesthat are used for both exposure analysis, such as style analysis, andthe simulation of portfolio returns. The additional asset classes,referred to as factors, represented in the factor model are conditionalupon the core asset class return scenarios generated by the pricingmodule 305. According to one embodiment, these additional factors maycorrespond to a set of asset classes or indices that are chosen in amanner to span the range of investments typically available toindividual investors in mainstream mutual funds and defined contributionplans. For example, the factors may be divided into the followinggroups: cash, bonds, equities, and foreign equities. The equities groupmay further be broken down into two different broad classifications (1)value versus growth and (2) market capitalization. Growth stocks arebasically stocks with relatively high prices relative to theirunderlying book value (e.g., high price-to-book ratio). In contrast,value stocks have relatively low prices relative to their underlyingbook value. With regard to market capitalization, stocks may be dividedinto groups of large, medium, and small capitalization. An exemplary setof factors is listed below in Table 1.

TABLE 1 Exemplary Set of Factors Group Factor Cash: Short Term US Bonds(core class) Bonds: Intermediate-term US Bonds (core class) Long-term USBonds (core class) US Corporate Bonds US Mortgage Backed SecuritiesNon-US Government Bonds Equities: Large Cap Stock -- Value Large CapStock -- Growth Mid Cap Stock -- Value Mid Cap Stock -- Growth Small CapStock -- Value Small Cap Stock -- Growth Foreign: International Equity-- Europe International Equity -- Pacific International Equity --Emerging Markets

At this point it is important to point out that more, less, or acompletely different set of factors may be employed depending upon thespecific implementation. The factors listed in Table 1 are simplypresented as an example of a set of factors that achieve the goal ofspanning the range of investments typically available to individualinvestors in mainstream mutual funds and defined contribution plans. Itwill be apparent to those of ordinary skill in the art that alternativefactors may be employed. In particular, it is possible to constructfactors that represent functions of the underlying asset classes forpricing of securities that are nonlinearly related to the prices ofcertain asset classes (e.g., derivative securities). In otherembodiments of the present invention, additional factors may be relevantto span a broader range of financial alternatives, such as industryspecific equity indices.

On a periodic basis, the financial product mapping module 315 mapsfinancial product returns onto the factor model. In one embodiment, theprocess of mapping financial product returns onto the factor modelcomprises decomposing financial product returns into exposures to thefactors. The mapping, in effect, indicates how the financial productreturns behave relative to the returns of the factors. According to oneembodiment, the financial product mapping module 315 is located on oneof the servers (e.g., the financial staging server 120, the broadcastserver 115, or the AdviceServer 110). In alternative embodiments, thefinancial product mapping module 315 may be located on the client 105.

In one embodiment of the present invention, an external approachreferred to as “returns-based style analysis” is employed to determine afinancial product's exposure to the factors. The approach is referred toas external because it does not rely upon information that may beavailable only from sources internal to the financial product. Rather,in this embodiment, typical exposures of the financial product to thefactors may be established based simply upon realized returns of afinancial product, as described further below. For more backgroundregarding returns-based style analysis see Sharpe, William F.“Determining a Fund's Effective Asset Mix,” Investment ManagementReview, December 1988, pp. 59-69 and Sharpe, William F. “AssetAllocation: Management Style and Performance Measurement,” The Journalof Portfolio Management, 18, no. 2 (Winter 1992), pp. 7-19 (“Sharpe[1992]”).

Alternative approaches to determining a financial product's exposure tothe factors include surveying the underlying assets held in a financialproduct (e.g. a mutual fund) via information filed with regulatorybodies, categorizing exposures based on standard industry classificationschemes (e.g. SIC codes), identifying the factors exposures based onanalysis of the structure of the product (e.g. equity index options, ormortgage backed securities), and obtaining exposure information based onthe target benchmark from the asset manager of the financial product. Ineach method, the primary function of the process is to determine the setof factor exposures that best describes the performance of the financialproduct.

The tax adjustment module 320 takes into account tax implications of thefinancial products and financial circumstances of the user. For example,the tax adjustment module 320 may provide methods to adjust taxableincome and savings, as well as estimates for future tax liabilitiesassociated with early distributions from pension and definedcontribution plans, and deferred taxes from investments in qualifiedplans. Further, the returns for financial products held in taxableinvestment vehicles (e.g. a standard brokerage account) may be adjustedto take into account expected tax effects for both accumulations anddistributions. For example, the component of returns attributable todividend income should be taxed at the user's income tax rate and thecomponent of returns attributable to capital gains should be taxed at anappropriate capital gains tax rate depending upon the holding period.

Additionally, the tax module 320 may forecast future components of thefinancial products total return due to dividend income versus capitalgains based upon one or more characteristics of the financial productsincluding, for example, the active or passive nature of the financialproduct's management, turnover ratio, and category of financial product.This allows precise calculations incorporating the specific tax effectsbased on the financial product and financial circumstances of theinvestor. Finally, the tax module 320 facilitates tax efficientinvesting by determining optimal asset allocation among taxable accounts(e.g., brokerage accounts) and nontaxable accounts (e.g., an IndividualRetirement Account (IRA), or employer sponsored 401(k) plan). In thismanner the tax module 320 is designed to estimate the tax impact for aparticular user with reference to that particular user's income taxrates, capital gains rates, and available financial products.Ultimately, the tax module 320 produces tax-adjusted returns for eachavailable financial product and tax-adjusted distributions for eachavailable financial product.

The portfolio optimization module 340 calculates the utility maximizingset of financial products under a set of constraints defined by the userand the available feasible investment set. In one embodiment, thecalculation is based upon a mean-variance optimization of the financialproducts. The constraints defined by the user may include bounds onasset class and/or specific financial product holdings. In addition,users can specify intermediate goals such as buying a house or putting achild through college, for example, that are incorporated into theoptimization. In any event, importantly, the optimization explicitlytakes into account the impact of future contributions and expectedwithdrawals on the optimal asset allocation. Additionally, thecovariance matrix used during optimization is calculated based upon theforecasts of expected returns for the factors generated by the factormodule 310 over the investment time horizon. As a result, the portfoliooptimization module 340 may explicitly take into account the impact ofdifferent investment horizons, including the horizon effects impact fromintermediate contributions and withdrawals.

The simulation processing module 330 provides additional analytics forthe processing of raw simulated return scenarios into statistics thatmay be displayed to the user via the UI 345. In the one embodiment ofthe present invention, these analytics generate statistics such as theprobability of attaining a certain goal, or the estimated time requiredto reach a certain level of assets with a certain probability. Thesimulation processing module 330 also incorporates methods to adjust thesimulated scenarios for the effects induced by sampling error inrelatively small samples. The simulation processing module 330 providesthe user with the ability to interact with the portfolio scenariosgenerated by the portfolio optimization module 340 in real-time.

The annuitization module 325 provides a meaningful way of representingthe user's portfolio value at the end of the term of the investmenthorizon. Rather than simply indicating to the user the total projectedportfolio value, one standard way of conveying the information to theuser is converting the projected portfolio value into a retirementincome number. The projected portfolio value at retirement may bedistributed over the length of retirement by dividing the projectedportfolio value by the length of retirement. More sophisticatedtechniques may involve determining how much the projected portfoliovalue will grow during retirement and additionally consider the effectsof inflation. In either event, however, these approaches erroneouslyassume the length of the retirement period is known in advance.

It is desirable, therefore, to present the user with a retirement incomenumber that is more representative of an actual standard of living thatcould be locked in for the duration of the user's retirement. Accordingto one embodiment, this retirement income number represents theinflation adjusted income that would be guaranteed by a real annuitypurchased from an insurance company or synthetically created via atrading strategy involving inflation-indexed treasury bond securities.In this manner, the mortality risk is taken out of the picture becauseregardless of the length of the retirement period, the user would beguaranteed a specific annual real income. To determine the retirementincome number, standard methods of annuitization employed by insurancecompanies may be employed. Additionally, mortality probabilities for anindividual of a given age, risk profile, and gender may be based onstandard actuarial tables used in the insurance industry. For moreinformation see Bowers, Newton L. Jr., et al, “Actuarial Mathematics,”The Society of Actuaries, Itasca, Ill., 1986, pp. 52-59 and Society ofActuaries Group Annuity Valuation Table Task Force, “1994 Group AnnuityMortality Table and 1994 Group Annuity Reserving Table,” Transactions ofthe Society of Actuaries, Volume XLVII, 1994, pp. 865-913. Calculatingthe value of an inflation-adjusted annuity value may involve estimatingthe appropriate values of real bonds of various maturities. The pricingmodule 305 generates the prices of real bonds used to calculate theimplied real annuity value of the portfolio at the investment horizon.

Referring now to the plan monitoring module 350, a mechanism is providedfor alerting the user of the occurrence of various predeterminedconditions involving characteristics of the recommended portfolio.Because the data upon which the portfolio optimization module 340depends is constantly changing, it is important to reevaluatecharacteristics of the recommended portfolio on a periodic basis so thatthe user may be notified in a timely manner when there is a need forhim/her to take affirmative action, for example. According to oneembodiment, the plan monitoring module 350 is located on theAdviceServer 110. In this manner, the plan monitoring module 350 hasconstant access to the user profile and portfolio data.

In one embodiment, the occurrence of two basic conditions may cause theplan monitoring module 350 to trigger a notification or alert to theuser. The first condition that may trigger an alert to the user is thecurrent probability of achieving a goal falling outside of apredetermined tolerance range of the desired probability of a achievingthe particular goal. Typically a goal is a financial goal, such as acertain retirement income or the accumulation of a certain amount ofmoney to put a child though college, for example. Additionally, the planmonitoring module 350 may alert the user even if the current probabilityof achieving the financial goal is within the predetermined tolerancerange if a measure of the currently recommended portfolio's utility hasfallen below a predetermined tolerance level. Various other conditionsare contemplated that may cause alerts to be generated. For example, ifthe nature of the financial products in the currently recommendedportfolio have changed such that the risk of the portfolio is outsidethe user's risk tolerance range, the user may receive an indication thathe/she should rebalance the portfolio. Plan monitoring processing,exemplary real world events that may lead to the above-described alertconditions, and additional alert conditions are described further below.

The UI module 345 provides mechanisms for data input and output toprovide the user with a means of interacting with and receiving feedbackfrom the financial advisory system 100, respectively. Furtherdescription of a UI that may be employed according to one embodiment ofthe present invention is disclosed in U.S. Pat. Nos. 5,918,217 and6,012,044, both entitled “USER INTERFACE FOR FINANCIAL ADVISORY SYSTEM,”the contents of which are hereby incorporated by reference.

Other modules may be included in the financial advisory system 100 suchas a pension module and a social security module. The pension module maybe provided to estimate pension benefits and income. The social securitymodule may provide estimates of the expected social security income thatan individual will receive upon retirement. The estimates may be basedon calculations used by the Social Security Administration (SSA), and onprobability distributions for reductions in the current level ofbenefits.

Core Asset Scenario Generation

FIG. 4 is a flow diagram illustrating core asset class scenariogeneration according to one embodiment of the present invention. Inembodiments of the present invention, core assets include short-term USgovernment bonds, long-term US government bonds, and US equities. Atstep 410, parameters for one or more functions describing statevariables are received. The state variables may include general economicfactors, such as inflation, interest rates, dividend growth, and othervariables. Typically, state variables are described by econometricmodels that are estimated based on observed historical data.

At step 420, these parameters are used to generate simulated values forthe state variables. The process begins with a set of initial conditionsfor each of the state variables. Subsequent values are generated byiterating the state variable function to generate new values conditionalon previously determined values and a randomly drawn innovation term. Insome embodiments, the state variable functions may be deterministicrather than stochastic. In general, the randomly drawn innovation termsfor the state variable functions may be correlated with a fixed orconditional covariance matrix.

At step 430, returns for core asset classes are generated conditional onthe values of the state variables. Returns of core asset classes may bedescribed by a function of a constant, previously determined core assetclass returns, previously determined values of the state variables, anda random innovation term. Subsequent values are generated by iterating acore asset class function to generate new values conditional onpreviously determined values and a random draws of the innovation term.In some embodiments, the core asset class functions may be deterministicrather than stochastic. In general, the randomly drawn innovation termsfor the core asset class functions may be correlated with a fixed orconditional covariance matrix.

In alternative embodiments, steps 410 and 420 may be omitted and thecore asset class returns may be generated directly in an unconditionalmanner. A simple example of such a model would be a function consistingof a constant and a randomly drawn innovation term.

A preferred approach would jointly generate core asset class returnsbased on a model that incorporates a stochastic process (also referredto as a pricing kernel) that limits the prices on the assets and payoffsin such a way that no arbitrage is possible. By further integrating adividend process with the other parameters an arbitrage free result canbe ensured across both stocks and bonds. Further description of such apricing kernel is disclosed in a copending U.S. patent applicationentitled “PRICING KERNEL FOR FINANCIAL ADVISORY SYSTEM,” applicationSer. No. 08/982,941, filed on Dec. 2, 1997, assigned to the assignee ofthe present invention, the contents of which are hereby incorporated byreference.

Factor Model Asset Scenario Generation

Referring now to FIG. 5, factor model asset scenario generation will nowbe described. A scenario in this context is a set of projected futurevalues for factors. According to this embodiment, the factors may bemapped onto the core asset factors by the following equation:

r_(ii)=α_(i)+β_(1i)ST_Bonds_(t)+β_(2i)LT_Bonds₁+β_(3i)US_Stocks₁+β₁  (EQ#1)

where

r_(it) represents the return for a factor, i, at time t

β_(ji) represent slope coefficients or the sensitivity of the factor ito core asset class j

ST_Bonds₁ is a core asset class representing the returns estimated bythe pricing module 305 for short-term US government bonds at time t

LT_Bonds is a core asset class representing the returns estimated by thepricing module 305 for long-term US government bonds at time t.

US_Stocks₁ is a core asset class representing the returns estimated bythe pricing module 305 for US stocks at time t.

α_(i) is a constant representing the average returns of factor assetclass i relative to the core asset class exposures (“factor alpha”).

ε₁ is a residual random variable representing the returns of factorasset class i that are not explained by the core asset class exposures(“residual variance”).

At step 510, the beta coefficients (also referred to as the loadings orslope coefficients) for each of the core asset classes are determined.According to one embodiment, a regression is run to estimate the valuesof the beta coefficients. The regression methodology may or may notinclude restrictions on the sign or magnitudes of the estimated betacoefficients. In particular, in one embodiment of the present invention,the coefficients may be restricted to sum to one. However, in otherembodiments, there may be no restrictions placed on the estimated betacoefficients.

Importantly, the alpha estimated by the regression is not used forgenerating the factor model asset scenarios. Estimates of alpha based onhistorical data are extremely noisy because the variance of the expectedreturns process is quite high relative to the mean. Based on limitedsample data, the estimated alphas are poor predictors of future expectedreturns. At any rate, according to one embodiment, a novel way ofestimating the alpha coefficients that reduces the probability ofstatistical error is used in the calibration of the factor model. Thisprocess imposes macroconsistency on the factor model by estimating thealpha coefficients relative to a known efficient portfolio, namely theMarket Portfolio. Macroconsistency is the property that expected returnsfor the factor asset classes are consistent with an observed marketequilibrium, that is estimated returns will result in markets clearingunder reasonable assumptions. The Market Portfolio is the portfoliodefined by the aggregate holdings of all asset classes. It is aportfolio consisting of a value-weighted investment in all factor assetclasses. Therefore, in the present example, macroconsistency may beachieved by setting the proportion invested in each factor equal to thepercentage of the total market capitalization represented by theparticular factor asset class.

At step 520, a reverse optimization may be performed to determine theimplied factor alpha for each factor based upon the holdings in theMarket Portfolio. This procedure determines a set of factor alphas thatguarantee consistency with the observed market equilibrium. In astandard portfolio optimization, Quadratic Programming (QP) is employedto maximize the following utility function:

$\begin{matrix}{{{{E(r)}^{T}X} - \frac{\left( {X^{T}{C(r)}X} \right)}{\tau}},{{u^{T}X} = 1}} & \left( {{EQ}\mspace{14mu} {\# 2}} \right)\end{matrix}$

where,

E(r) represents expected returns for the asset classes,

C(r) represents the covariance matrix for the asset class returns,

τ, Tau, represents a risk tolerance value,

X is a matrix representing the proportionate holdings of each assetclass of an optimal portfolio comprising the asset classes, and

u is a vector of all ones.

C(r) may be estimated from historical returns data or moreadvantageously may be estimated from projected returns generated by apricing kernel model.

Inputs to a standard portfolio optimization problem include E(r), C(r),and Tau and QP is used to determine X. However, in this case, X is givenby the Market Portfolio, as described above, and a reverse optimizationsolves for E(r) by simply backing out the expected returns that yield Xequal to the proportions of the Market Portfolio.

Quadratic Programming (QP) is a technique for solving an optimizationproblem involving a quadratic (squared terms) objective function withlinear equality and/or inequality constraints. A number of different QPtechniques exist, each with different properties. For example, some arebetter for suited for small problems, while others are better suited forlarge problems. Some are better for problems with very few constraintsand some are better for problems with a large number of constraints.According to one embodiment of the present invention, when QP is calledfor, an approach referred to as an “active set” method is employedherein. The active set method is explained in Gill, Murray, and Wright,“Practical Optimization,” Academic Press, 1981, Chapter 5.

The first order conditions for the optimization of Equation #2 are:

$\begin{matrix}{{E(r)} = {{2\; {C(r)}\frac{X}{\tau}} + {K\; u}}} & \left( {{EQ}\mspace{14mu} {\# 3}} \right)\end{matrix}$

where K is a Lagrange multiplier; hence, knowing the Market Portfolioand any two values of E(r) (for example, the risk free rate and thereturn on US equities) the full set of expected returns that areconsistent with the Market Portfolio can be derived. The two values ofE(r) required for the reverse optimization follow from the expectedreturns of the core assets.

At step 530, factor returns may be generated based upon the estimatedalphas from step 520 and the estimated beta coefficients from step 510.As many factor model asset scenarios as are desired may be generatedusing Equation #1 and random draws for the innovation value ε₁. A randomvalue for ε₁, is selected for each evaluation of Equation #1. Accordingto one embodiment, ε₁, is distributed as a standard normal variate. Inother words ε₁ is drawn from a standard normal distribution with a meanof 0 and a standard deviation of 1.

Advantageously, in this manner, a means of simulating future economicscenarios and determining the interrelation of asset classes isprovided.

Financial Product Exposure Determination

As discussed above, one method of determining how a financial productbehaves relative to a set of factor asset classes is to performreturns-based style analysis. According to one embodiment, returns for agiven financial product may be estimated as a function of returns interms of one or more of the factor asset classes described above basedon the following equation:

r _(ft)=α_(ft) +S _(f1) r ₁ t+S _(f2) r ₂ t+ . . . +S _(fn) r_(nt)+ε_(t)  (EQ #4)

where,

α_(ft) is the mean of the left over residual risk (“selection variance”)of the financial product return that cannot be explained in terms of thefactor loadings.

r_(ft) is the return for financial product f at time t,

r_(nt) is the return for factor n at time t, and

ε_(t) is the residual at time t that is unexplained by movements in thefactor returns.

The financial product exposure determination module 315 computes thefactor asset class exposures for a particular fund via a nonlinearestimation procedure. The exposure estimates, S_(fn), are called stylecoefficients, and are generally restricted to the range [0,1] and to sumto one. In other embodiments, these restrictions may be relaxed (forexample, with financial products that may involve short positions, thecoefficients could be negative). Alpha may be thought of as a measure ofthe relative under or over performance of a particular fund relative toits passive style benchmark.

At this point in the process, the goal is to take any individual groupof assets that people might hold, such as a group of mutual funds, andmap those assets onto the factor model, thus allowing portfolios to besimulated forward in time. According to one embodiment, this mapping isachieved with what is referred to as “returns-based style analysis” asdescribed in Sharpe [1992], which is hereby incorporated by reference.Generally, the term “style analysis” refers to determining a financialproduct's exposure to changes in the returns of a set of major assetclasses using Quadratic Programming or similar techniques.

FIG. 6 is a flow diagram illustrating a method of determining afinancial product's exposures to factor asset class returns according toone embodiment of the present invention. At step 610, the historicalreturns for one or more financial products to be analyzed are received.According to one embodiment, the financial product exposure module 315may reside on a server device and periodically retrieve the historicalreturn data from a historical database stored in another portion of thesame computer system, such as RAM, a hard disk, an optical disc, orother storage device. Alternatively, the financial product exposuremodule 325 may reside on a client system and receive the historicalreturn data from a server device as needed. At step 620, factor assetclass returns are received.

At step 630, QP techniques or the like are employed to determineestimated exposures (the S coefficients) to the factor asset classreturns.

At step 640, for each financial product, expected future alpha isdetermined for each subperiod of the desired scenario period. Withregards to mutual finds or related financial products, for example,historical alpha alone is not a good estimate of future alpha. That is,a given mutual fund or related financial product will not continue tooutperform/under perform its peers indefinitely into the future. Rather,empirical evidence suggests that over performance may partially persistover one to two years while under performance may persist somewhatlonger (see for example, Carhart, Mark M. “On Persistence in Mutual FundPerformance.” Journal of Finance, March 1997, Volume 52 No. 1, pp.57-82).

For example, future alpha may depend upon a number of factors, such asturnover, expense ratio, and historical alpha. Importantly, one or moreof these factors may be more or less important for particular types offunds. For example, it is much more costly to buy and sell in emergingmarkets as compared to the market for large capitalization US equities.In contrast, bond turnover can be achieved at a much lower cost,therefore, turnover has much less affect on the future alpha of a bondfund than an equity fund. Consequently, the penalty for turnover may behigher for emerging market funds compared to large cap U.S. equities andbond funds. Improved results may be achieved by taking into accountadditional characteristics of the fund, such as the fact that the fundis an index fund and the size of the fund as measured by total netassets, for example.

According to one embodiment of the present invention, a moresophisticated model is employed for determining future alpha for eachfund:

α_(t) =a _(base)+ρ^(t)(α_(historical)−α_(base))  (EQ #5)

where,

a_(base) is the baseline prediction for future Alpha of the fund

ρ, Rho, governs the speed of decay from α_(historical) to α_(base)

a_(historical) is Alpha estimated in Equation #4

According to one embodiment,

α_(base) =C+β ₁Expense_Ratio+β₂Turnover+β₃Fund_Size  (EQ #6)

where the parameters are estimated separately for each of four differentclasses of funds: US equity, foreign equity, taxable bond, nontaxablebond. These parameters may be estimated using conventional econometrictechniques, such as ordinary least squares (OLS). According to oneembodiment, Rho is estimated by first calculating historical deviationsfrom α_(base) (“residual alpha”) and then estimating Rho as the firstorder serial correlation of the residual alpha series.

Portfolio Optimization

Portfolio optimization is the process of determining a set of financialproducts that maximizes the utility function of a user. According to oneembodiment, portfolio optimization processing assumes that users have amean-variant utility function, namely, that people like having morewealth and dislike volatility of wealth. Based on this assumption andgiven a user's risk tolerance, the portfolio optimization module 340calculates the mean-variance efficient portfolio from the set offinancial products available to the user. As described above,constraints defined by the user may also be taken into consideration bythe optimization process. For example, the user may indicate a desire tohave a certain percentage of his/her portfolio allocated to a particularfinancial product. In this example, the optimization module 340determines the allocation among the unconstrained financial productssuch that the recommended portfolio as a whole accommodates the user'sconstraint(s) and is optimal for the user's level of risk tolerance.

Prior art mean-variant portfolio optimization traditionally treats theproblem as a single period optimization. Importantly, in the embodimentsdescribed herein, the portfolio optimization problem is structured insuch as way that it may explicitly take into account the impact ofdifferent investment horizons and the impact of intermediatecontributions and withdrawals. Further the problem is set up so that itmay be solved with QP methods.

Referring now to FIG. 7, a method of portfolio optimization according toone embodiment of the present invention will now be described. At step710, information regarding expected withdrawals is received. Thisinformation may include the dollar amount and timing of the expectedwithdrawal. At step 720, information regarding expected futurecontributions is received. According to one embodiment, this informationmay be in the form of a savings rate expressed as a percentage of theuser's gross income or alternatively a constant or variable dollar valuemay be specified by the user.

At step 730, information regarding the relevant investment time horizonis received. In an implementation designed for retirement planning, forexample, the time horizon might represent the user's desired retirementage.

At step 740, information regarding the user's risk tolerance, Tau, isreceived.

At step 750, the mean-variance efficient portfolio is determined.According to one embodiment, wealth in real dollars at time T isoptimized by maximizing the following mean-variance utility function bydetermining portfolio proportions (X_(i)):

$\begin{matrix}{U = {{E\left( W_{T} \right)} - \frac{{Var}\left( W_{T} \right)}{\tau}}} & \left( {{EQ}\mspace{14mu} {\# 7}} \right)\end{matrix}$

where for a given scenario,

E(W_(T)) is the expected value of wealth at a time T

Var(W_(T)) is the variance of wealth at time T

τ is the user's risk tolerance

$\begin{matrix}{W_{T} = {{X_{1}{\sum\limits_{t = 0}^{T - 1}\; {C_{t}{\prod\limits_{j = {t + 1}}^{T}\; \left( {1 + R_{j\; 1}} \right)}}}} + \ldots + {X_{n}{\sum\limits_{t = 0}^{T - 1}\; {C_{t}{\prod\limits_{j = {t + 1}}^{T}\; \left( {1 + R_{jn}} \right)}}}} + g}} & \left( {{EQ}\mspace{14mu} {\# 8}} \right)\end{matrix}$

where,

X_(i) represents the recommended constant proportion of each netcontribution that should be allocated to financial product i.

C_(t) represents the net contribution at time t,

R_(ji) represents the expected returns for financial product i in yearj,

n is the number of financial products that are available foroptimization,

g is the value of constrained assets for a given scenario,

The product of gross returns represents the compounding of values fromyear 1 to the horizon. Initial wealth in the portfolio is represented bycontribution C₀.

Importantly, the financial product returns need not represent fixedallocations of a single financial product. Within the context of theoptimization problem, any individual asset return may be composed of astatic or dynamic strategy involving one or more financial products. Forexample, one of the assets may itself represent a constant re-balancedstrategy over a group of financial products. Moreover, any dynamicstrategy that can be formulated as an algorithm may be incorporated intothe portfolio optimization. For example, an algorithm which specifiesrisk tolerance which decreases with the age of the user could beimplemented. It is also possible to incorporate path dependentalgorithms (e.g., portfolio insurance).

According to Equation #8, contributions are made from the current yearto the year prior to retirement. Typically, a contribution made at timet will be invested from time t until retirement. An exception to thiswould be if a user specifies a withdrawal, in which case a portion ofthe contribution may only be held until the expected withdrawal date.

An alternative to the buy and hold investment strategy assumed abovewould be to implement a “constant mix” investment strategy orre-balancing strategy. For purposes of this example, it is assumed thatthe recommended fixed target asset-mix will be held in an account foreach year in the future. Therefore, each year, assets will be boughtand/or sold to achieve the target. Let f_(i) be the fraction of accountwealth targeted for the i-th asset, then the sum of the fractions mustequal one.

In the following “evolution” equations, nominal wealth aggregation ismodeled for a single taxable account from the current time t=0 to thetime horizon t=T. It is assumed that “N” assets are in the account,labeled by the set of subscripts {i=1, . . . , N}. The superscriptsminus and plus are used to distinguish between the values of a variablejust before, and just after, “settlement”. The settlement “event”includes paying taxes on distributions and capital gains, investing newcontributions, buying and selling assets to achieve the constant mix,and paying load fees. For example, W⁺(t) is the total wealth invested inall assets just after settlement at time “t”. The evolution equationsfor the pre- and post-settlement values, the “dollars” actually investedin each asset, are:

$\begin{matrix}{{W_{i}^{-}(t)} = \left\{ \begin{matrix}{{W_{i}^{-}(0)},} & {{t = 0},} \\{{{\left\lbrack {1 + {R_{i}(t)}} \right\rbrack \cdot {W_{i}^{+}\left( {t - 1} \right)}} - {{k_{i}(t)}}},} & {{0 < t \leq T},}\end{matrix} \right.} & \left( {19\; a} \right) \\{{W_{i}^{+}(t)} = \left\{ \begin{matrix}{{f_{i} \cdot {W^{+}(t)}},} & {{0 \leq t},T,} \\{0,} & {t = {T.}}\end{matrix} \right.} & \left( {19\; b} \right)\end{matrix}$

In the above equation, the double-bar operator ∥ ∥ is equal to eitherits argument or zero, whichever is greater. From Eq. (19a), we see thatthe pre-settlement value at any time (after the initial time) is justthe gross return on the post-settlement value of the previous time lessthe “positive-part” of any distribution, i.e. the “dividend”. Here,k_(i)(t) is the portion of the return of the i-th asset that isdistributed, and R_(i)(t) is the total nominal return on the i-th assetin the one-year period [t−1, t]. We also assume that an initial,pre-settlement value is given for each asset. Eq. (19b) defines thepost-settlement value in terms of the asset's constant mix and the totalaccount value after settlement. Since we “cash-out” the portfolio at thetime horizon, the final amount in each asset at t=T is zero. The pre-and post-settlement, total values are governed by the pair of equations:

$\begin{matrix}{{{W^{-}(t)} = {\sum\limits_{i = 1}^{N}\; {W_{i}^{-}(t)}}},\mspace{14mu} {0 \leq t \leq T},} & \left( {19\; c} \right) \\{{{W^{+}(t)} = {{W^{-}(t)} + {C(t)} + {D(t)} - {L(t)} - {S(t)}}},\mspace{14mu} {0 \leq t \leq {T.}}} & \left( {19\; d} \right)\end{matrix}$

In Eq. (19d), C(t) is the nominal contribution to the account at time“t”, D(t) is the total of all distributed “dividends”, L(t) is the“leakage”, the total amount paid in loads to both rebalance and toinvest additional contributions, and S(t) is the “shrinkage”, the totalamount paid in taxes on distributions and capital gains. We note thatW⁺(T) is the final horizon wealth after all taxes have been paid. Thevalue of D(t), the total of all distributed dividends, is the sum of thepositive distributions:

$\begin{matrix}{{{D(t)} = {\sum\limits_{i = 1}^{N}\; {{k_{i}(t)}}}},\mspace{14mu} {0 \leq t \leq {T.}}} & \left( {19\; e} \right)\end{matrix}$

Similarly, the “leakage” L(t) is the total amount of dollars paid inloads, and L_(i)(t) is the number of dollars paid in loads on just thei-th asset. These individual loads depend on l_(i), the front-end loadfee (a rate) on the i-th asset.

$\begin{matrix}{{L_{i} = {\left\lbrack {l_{i}/\left( {1 - l_{i}} \right)} \right\rbrack \cdot {{{W_{i}^{+}(t)} - {{k_{i}(t)}} - {W_{i}^{-}(t)}}}}},\mspace{14mu} {0 \leq t \leq {T.}}} & \left( {19\; f} \right) \\{{{L(t)} = {\sum\limits_{i = 1}^{N}\; {L_{i}(t)}}},\mspace{14mu} {0 \leq t \leq {T.}}} & \left( {19\; g} \right)\end{matrix}$

If there is a short-term loss (negative distribution), the load fee paidon an asset's purchase is just a fixed fraction of the purchaseprice.^(i) When there is a short-term gain (positive distribution), wecan re-invest any part of it without load fees, and pay fees only onpurchases in excess of the gain. Note that at the horizon, we“cash-out”, and don't pay any load fees. The dollar amount of a load feeis proportional to the ratio l/(1−l). That's because our wealthvariables are all measured as “net” loads. To see this, suppose we makea contribution c. After loads, we are left with W=(1−l) c. In terms ofW, the amount we paid in loads is L=lc=[l/(1−l)] W.

The equation for the “shrinkage” S(t), the total amount paid in taxes,has two terms. The first term is the tax on distributions and ismultiplied by the marginal tax-rate; the second term is the tax oncapital gains and is multiplied by the capital gains tax-rate.

$\begin{matrix}{{{{S(t)} = {{\tau_{m} \cdot {\sum\limits_{i = 1}^{N}{k_{i}(t)}}} + {\tau_{cg} \cdot {\sum\limits_{i = 1}^{N}{\left\lbrack {1 - {{B_{i}\left( {t - 1} \right)}/{W_{i}^{-}(t)}}} \right\rbrack \cdot {{{W_{i}^{-}(t)} - {W_{i}^{+}(t)}}}}}}}},{0 \leq t \leq {T.}}}\mspace{14mu}} & \left( {19\; h} \right)\end{matrix}$

In Eq. (19h), the capital gains tax depends on the basis B_(i)(t), thetotal of all after-tax nominal-dollars that have been invested in thei-th asset up to time “t”. Note that there can be either a capital gainor loss. The double-bar operator ensures that capital gains aretriggered only when there is a sale of assets. At the horizon, we sellall assets, and automatically pay all taxes. The basis B_(i)(t), evolvesaccording to the following recursion equation:

$\begin{matrix}{{B_{i}(t)} = \left\{ \begin{matrix}{{B_{i}(0)},} & {{t = 0},} \\{{B_{i}\left( {t - 1} \right)} + {{{W_{i}^{+}(t)} - {W_{i}^{-}(t)}}} + {L_{i}(t)}} & \; \\{{{- \left\lbrack {{B_{i}\left( {t - 1} \right)}/{W_{i}^{-}(t)}} \right\rbrack} \cdot {{{W_{i}^{-}(t)} - {W_{i}^{+}(t)}}}},} & {0 \leq t \leq {T.}}\end{matrix} \right.} & \left( {19\; i} \right)\end{matrix}$

Note that all new purchases are made with after-tax dollars, and add tothe basis; all sales decrease the basis. Further, any load paid topurchase an asset adds to the basis. We assume that the initial basisB_(i)(0) of an asset is either given, or defaults to the initial,pre-settlement value so that the average basis is initially equal toone.

A “constitutive” equation for k_(i)(t) is needed to complete our systemof equations. Short-term distributions depend on the “type” of asset;here we model mutual funds:

$\begin{matrix}{{k_{i}(t)} = \left\{ \begin{matrix}{{k_{i}(0)},} & {{t = 0},} \\{{\kappa_{i} \cdot {R_{i}(t)} \cdot {W_{i}^{+}\left( {t - 1} \right)}},} & {0 < t \leq {T.}}\end{matrix} \right.} & \left( {20\; a} \right)\end{matrix}$

Often, we set the initial distribution to zero, and assume that theasset's initial pre-settlement value has already accounted for anynon-zero, initial value. We note that the distribution is proportionalto the amount of wealth at “stake” during the prior-period. For mutualfunds, we assume that the distribution is a fraction κ_(i) of theprior-period's total return, and therefore is also proportional toR_(i)(t). Note that the distribution in Eq. (20a) can be a gain(positive) or a loss (negative). In contrast, the constitutive equationfor stocks takes the form:

$\begin{matrix}{{k_{i}(t)} = \left\{ \begin{matrix}{{k_{i}(0)},} & {{t = 0},} \\{{\kappa_{i} \cdot \left\lbrack {1 + {R_{i}(t)}} \right\rbrack \cdot {W_{i}^{+}\left( {t - 1} \right)}},} & {0 < t \leq {T.}}\end{matrix} \right.} & \left( {20\; b} \right)\end{matrix}$

For stocks, the proportionality constant κ_(i) models a constantdividend “yield”, and the distribution is always a gain (non-negative).For stocks (mutual funds), the distribution is proportional to the gross(simple) return.

Before we leave this section, a word on 401(k) plans and IRA's (with noload funds). For these accounts, the loads and taxes are ignored, andthere is no basis in the asset. At “settlement”, the user justre-balances their account. The evolution equations for these accounts istrivial in comparison to the equations for a general taxable account:

$\begin{matrix}{{{W_{i}^{+}(t)} = {f_{i} \cdot {W^{+}(t)}}},\mspace{14mu} {0 \leq t \leq T},} & \left( {21\; a} \right) \\{{W^{+}(t)} = \left\{ \begin{matrix}{{W^{+}(0)},} & {{t = 0},} \\\begin{matrix}{\left( {1 + {\sum\limits_{i = 1}^{N}{f_{i} \cdot {R_{i}(t)}}}} \right) \cdot} \\{{{W^{+}\left( {t - 1} \right)} + {C(t)}},}\end{matrix} & {0 < t \leq {T.}}\end{matrix} \right.} & \left( {21\; b} \right)\end{matrix}$

At the time horizon T, the total wealth in a non-taxable account is justW⁺(T). This is a pre-withdrawal total value. When retirement withdrawalsare made from a tax-free account, they are taxed at the client's averagetax-rate, τ_(a). Therefore, the “after-tax” equivalent value is equal to“pre-tax” wealth W⁺(T) times the tax factor (1−τ_(a)).

How do we aggregate taxable and non-taxable accounts to get totalportfolio wealth? We choose non-taxable accounts as a baseline. If allthe funds in a non-taxable account were converted to an annuity, and theannuity payments were taken as withdrawals, then the withdrawals wouldmimic a salary subject to income taxes. This is precisely the client'spre-retirement situation. Before aggregating a taxable account, we scaleits “after-tax” value to this baseline using the formula:

W_(baseline)=W_(after-tax)/(1−τ_(a))  (22)

Essentially, the baseline equivalent is obtained by grossing up valuesusing the average tax-rate.

The evolution equation variables appear “implicitly” in the recursionrelations. Hence, we need to “iterate” at each time step to solve for“explicit” variable values.^(ii) We illustrate this process with anexample. Consider the simple case where there are no distributions,contributions, or taxes; just loads, and a constant-mix strategy. Here,the evolution equations simplify to a single equation for the total,after-settlement wealth W⁺(t): ^(ii) In practice a robust root-findingalgorithm may be used rather than iteration.

$\begin{matrix}{{W^{+}(t)} = {{{W^{+ \cdot}\left( {t - 1} \right)} \cdot {\sum\limits_{i = 1}^{N}{f_{i} \cdot \left\lbrack {1 + {R_{i}(t)}} \right\rbrack}}} - {\sum\limits_{i = 1}^{N}{f_{i} \cdot \left\lbrack {l_{i}/\left( {1 - l_{i}} \right)} \right\rbrack \cdot {{{{W^{+}(t)} - {\left\lbrack {1 + {R_{i}(t)}} \right\rbrack \cdot {W^{+}\left( {t - 1} \right)}}}}.}}}}} & (23)\end{matrix}$

Note, we only know W⁺(t) as an implicit function of W⁺(t−1), but given aguess for it's value, we can refine the guess by substituting it intothe right-side of Eq. (23).

It's instructive to re-write Eq. (23) as the pair of equations in termsof an “effective” return R_(e)(t):

$\begin{matrix}{{{W^{+}(t)} = {\left\lbrack {1 + {R_{e}(t)}} \right\rbrack \cdot {W^{+}\left( {t - 1} \right)}}},} & \left( {24a} \right) \\{{R_{e}(t)} = {{\sum\limits_{i = 1}^{N}\; {f_{i} \cdot {R_{i}(t)}}} - {\sum\limits_{i = 1}^{N}\; {f_{i} \cdot \left\lbrack {l_{i}/\left( {1 - l_{i}} \right)} \right\rbrack \cdot {{{{R_{e}(t)} - {R_{i}(t)}}}.}}}}} & \left( {24b} \right)\end{matrix}$

Eq. (24a) is the evolution equation for a single asset with theeffective return. Eq. (24b) is an implicit equation for the effectivereturn R (t) in terms of the asset returns R_(i)(t). We solve for theeffective return using iteration. When the loads are equal to zero, asexpected, the effective return is just a weighted-average of the assetreturns. Even when the loads are not zero, this average return is a goodinitial guess for the iteration procedure. In fact, using the averagereturn as the initial guess and iterating once yields the followingexplicit approximation for the effective return:

$\begin{matrix}{{{R_{wgt}(t)} = {\sum\limits_{i = 1}^{N}\; {f_{i} \cdot {R_{i}(t)}}}},} & \left( {25a} \right) \\{{R_{e}(t)} \approx {{R_{wgt}(t)} - {\sum\limits_{i = 1}^{N}\; {f_{i} \cdot l_{i} \cdot {{{{R_{wgt}(t)} - {R_{i}(t)}}}.}}}}} & \left( {25b} \right)\end{matrix}$

Eq. (25b) is consistent with our intuition, and agrees well with higherorder iterates.

To determine the mutual fund input moments we must first calculate thekernel moments. This procedure calculates successive annual kernelmoments and averages the result. The resulting mean and covariancematrix is then utilized by the reverse optimization procedure and alsoas an input into the optimization procedure.

To calculate analytic core moments, first we must describe the wealthfor each core asset for an arbitrary holding period. For each of thecore assets, the resulting wealth from an arbitrary investment horizoncan be written as: [Note, this is an approximation for equities]

$W_{t,T} = {\exp \left\{ {{\sum\limits_{j = t}^{T - 1}\; a} + {bX}_{j + 1} + {c{\prod\limits_{j + 1}\; {{+ d}\; \delta_{j + 1}}}} + {eX}_{j} + {f{\prod\limits_{j}\; {{+ g}\; \delta_{j}}}}} \right\}}$

Where:

a, b, c, d, e, f, g=Constants

-   -   X_(j)=Real rate in year j    -   Π=Inflation rate in year j    -   δ_(j)=Dividend growth rate in year j

The expectation of wealth for any of the core assets given informationat time zero is then:

${E_{0}W_{t,T}} = {^{a{({T - t})}}E_{0}^{{\sum\limits_{j = t}^{T - 1}\; {eX}_{j}} + {bX}_{j + 1}}E_{0}^{\sum\limits_{j = t}^{T - 1}\; {f\; {\prod\limits_{j}\; {{+ c}\; \prod\limits_{j + 1}}}}}E_{0}^{{\sum\limits_{j = t}^{T - 1}\; {g\; \delta_{j}}} + {d\; \delta_{j + t}}}}$

Since X, Π, and δ are independent, we can deal with each of theseexpectations separately. For example, consider the contribution in theabove equation from inflation. The summation can be rewritten as:

${E_{0}\exp \left\{ {\sum\limits_{j = t}^{T - 1}\; {f\; {\prod\limits_{j}\; {{+ c}\; \prod\limits_{j + 1}}}}}\; \right\}} = {E_{0}\exp \left\{ \begin{matrix}{f\; {\prod\limits_{t}\; +}} \\{\left( {\sum\limits_{j = {t + 1}}^{T - 1}\; {\left( {f + c} \right)\prod\limits_{j}}}\; \right) + {c\; \prod\limits_{T}}}\end{matrix}\; \right\}}$

Next, we need to use iterated expectations to determine thisexpectation. We can write the expectation at time zero as the repeatedexpectation over the various innovations. For example, the equation forinflation can be rewritten as:

${E_{0}\exp \begin{Bmatrix}{f\; {\prod\limits_{t}\; +}} \\{\left( {\sum\limits_{j = {t + 1}}^{T - 1}\; {\left( {f + c} \right)\prod\limits_{j}}}\; \right) + {c\prod\limits_{T}}}\end{Bmatrix}} = {{E_{ɛ_{1}}E_{ɛ_{2}}{\ldots E}_{ɛ_{T}}\exp \begin{Bmatrix}{f\; {\prod\limits_{t} +}} \\{\left( {\sum\limits_{j = {t + 1}}^{T - 1}{\left( {f + c} \right)\prod\limits_{j}}} \right) + {c\prod\limits_{T}}}\end{Bmatrix}} = {E_{ɛ_{1}}E_{ɛ_{2}}\ldots \mspace{14mu} E_{ɛ_{T - 1}}\exp \begin{Bmatrix}{f\; {\prod\limits_{t} +}} \\\left( {\sum\limits_{j = {t + 1}}^{T - 1}\; {\left( {f + c} \right)\prod\limits_{j}}}\; \right)\end{Bmatrix}{E_{ɛ_{T}}\left\lbrack ^{c\prod\limits_{T}} \right\rbrack}}}$

Assuming inflation follows a modified square root process:

Π_(t)=μ_(π)=ρ_(π)Π_(t−1)+σ_(π)√{square root over (∥Π_(t−1)∥)}ε_(t)

Where ∥ ∥ denotes the Heaviside function

${\prod\limits_{t}\; } \equiv \left\{ \begin{matrix}0 & {{{if}\mspace{14mu} \prod\limits_{t}}\; \leq 0} \\{\prod\limits_{t}\;} & {{{if}\mspace{14mu} \prod\limits_{t}}\; > 0}\end{matrix} \right.$

Now we recursively start taking the expectations over epsilon startingat the end and working backward. So:

${E_{ɛ_{T}}\left\lbrack ^{c\; \prod\limits_{T}} \right\rbrack} = {{E_{ɛ_{T}}\left\lbrack ^{{c\; \mu_{\pi}} + {c\; \rho_{\pi}{\prod\limits_{T - 1}\; {{+ c}\; \sigma_{\pi}\sqrt{\prod\limits_{T - 1}\; }ɛ_{T}}}}} \right\rbrack} \approx ^{c(\begin{matrix}{\mu_{\pi} + {\rho_{\pi}{\prod\limits_{T - 1}\; +}}} \\{\frac{1}{2}c\; \sigma_{\pi}^{2}\prod\limits_{T - 1}}\end{matrix})}}$

Where the approximation is due to the Heaviside function.

Combining this with the above equation yields:

${E_{ɛ_{1}}E_{ɛ_{2}}{\ldots E}_{ɛ_{T - 1}}\exp \begin{Bmatrix}{f\; {\prod\limits_{t} +}} \\\left( {\sum\limits_{j = {t + 1}}^{T - 1}{\left( {f + c} \right)\prod\limits_{j}}} \right)\end{Bmatrix}{E_{ɛ_{T}}\left\lbrack ^{c\prod\limits_{T}} \right\rbrack}} = {E_{ɛ_{1}}E_{ɛ_{2}}\ldots \mspace{14mu} E_{ɛ_{T - 2}}\exp \begin{Bmatrix}{f\; {\prod\limits_{t} +}} \\\left( {\sum\limits_{j = {t + 1}}^{T - 1}\; {\left( {f + c} \right)\prod\limits_{j}}}\; \right)\end{Bmatrix}{E_{ɛ_{T - 1}}\left\lbrack ^{{c\; \mu_{\pi}} + {{(\begin{matrix}{{c\; \rho_{\pi}}\; + {\frac{1}{2}c^{2}}} \\{\; {\sigma_{\pi}^{2} + c + f}}\end{matrix})}\prod\limits_{T - 1}}} \right\rbrack}}$

In general for any time period t, an exponential linear function of Πhas the following expectation:

$\begin{matrix}{{E_{ɛ_{t}}\left\lbrack ^{A_{j} + {B_{j}\prod\limits_{t}}} \right\rbrack} = {E_{ɛ_{t}}\left\lbrack ^{A_{j} + {B_{j}{({\mu_{\pi} + {\rho_{\pi}{\prod\limits_{t - 1}\; {{+ \sigma_{\pi}}{\prod\limits_{t - 1}\; }ɛ_{t}}}}})}}} \right\rbrack}} \\{= ^{A_{j} + {B_{j}\mu_{\pi}} + {B_{j}{\prod\limits_{t - 1}{({\rho_{\pi} + {\frac{1}{2}\sigma_{\pi}^{2}B_{j}}})}}}}} \\{= ^{A_{j} + {B_{j}\mu_{\pi}} + {{({B_{j}\; {({\rho_{\pi} + {\frac{1}{2}\sigma_{\pi}^{2}B_{j}}})}})}\prod\limits_{t - 1}}}} \\{= ^{A_{j - 1} + {B_{j - 1}\prod\limits_{t - 1}}}}\end{matrix}$

The critical feature is that an exponential linear function of IIremains exponential linear after taking the expectation. This invarianceallows for the backward recursion calculation. Only the constant (A) andthe slope (B) are changing with repeated application of the expectationoperator. The evolution of A and B can be summarized as

A _(J) =A _(J+1)+μ_(π) B _(J+1)

B _(J) =B _(J+1)[ρ_(π)+1/2σ_(π) ² B _(J+1)]

In addition, the B_(j) coefficient has to be increased by (c+f) toaccount for the additional Π_(j) term in the summation. To implementthis recursive algorithm to solve for expected wealth, first define thefollowing indicator variable:

${I\left( {t_{1},t_{2}} \right)} = \begin{Bmatrix}1 & {{{if}\mspace{14mu} t_{1}} \leq j \leq t_{2}} \\0 & {Otherwise}\end{Bmatrix}$

Next, the following algorithm may be employed:

InitialConditions J=T,A_(T)=0,B_(T)=c

J=J−1  (1)

A _(J) =A _(J+1)+μ_(π) B _(J+1)

B _(J) =B _(J+1)[ρ_(π)+1/2σ_(π) ² B _(J+1)]+c·I(t+1,T−1)+f·I(t,T−1)  (2)

if J=0,End

E(W _(t,T))=e ^(A) ¹ _(+B) ¹ ^(n) ⁰   (3)

Go To (1)  (4)

The same technique applies to X since it is also a square root process.A similar technique can be used to create a recursive algorithm for theδ component. The only difference is that δ is an AR(1) process insteadof a square root process.

In particular,

For this AR(1) process, the expectation is of the following form.

$\begin{matrix}{{E_{ɛ_{t}}\left\lbrack ^{A_{j} + {B_{j}\delta_{t}}} \right\rbrack} = {E_{ɛ_{T}}\left\lbrack ^{A_{j} + {B_{j}{({\mu_{\delta} + {\rho_{\delta}\delta_{t - 1}} + {\sigma_{\delta}ɛ_{t}}})}}} \right\rbrack}} \\{= ^{A_{j} + {B_{j}\mu_{\delta}} + {\frac{1}{2}\sigma_{\pi}^{2}B_{j}} + {B_{j}\rho_{\delta}\delta_{t - 1}}}} \\{= ^{A_{j - 1} + {B_{j - 1}\delta_{t - 1}}}}\end{matrix}$

The evolution of A and B is thus summarized as:

A _(J) =A _(J+1) +B _(J+1)(μ_(δ)+1/2σ_(δ) ²)

B _(J) =B _(J+1)ρ_(δ)

The recursive relationship for δ is then:

InitialConditions J=T,A_(T)=0,B_(T)=d

J=J−1  (1)

A _(J) =A _(J+1) +B _(J+1)(μ_(δ)+1/2σ_(δ) ²)  (2)

B _(J) =B _(J+1)ρ_(δ) +d·I(t+1,T−1)+g·I(t,T−1)

if J=0,End

E(W _(t,T))=e ^(A) ¹ ^(+B) ¹ ^(δ) ⁰   (3)

Go To (1)  (4)

This framework for calculating expected wealth can also be used tocalculate the variance of wealth for an arbitrary holding period. Fromthe definition of variance, we have:

V₀(W_(t, T)) = E₀(W_(t, T)²) − E₀(W_(t, T))² but $\begin{matrix}{W_{t,T}^{2} = \left\lbrack {\exp \left\{ {{\sum\limits_{j = t}^{T - 1}\; {a \cdot {bX}_{j + 1}}} + {c{\prod\limits_{j + 1}\; {{+ d}\; \delta_{j + 1}}}} + {eX}_{j} + {f{\prod\limits_{j}\; {{+ g}\; \delta_{j}}}}} \right\}} \right\rbrack^{2}} \\{= {\exp \left\{ {\sum\limits_{j = t}^{T - 1}\; {2\left( {a + {bX}_{j + 1} + {c{\prod\limits_{j + 1}\; {{+ d}\; \delta_{j + 1}}}} + {eX}_{j} + {f{\prod\limits_{j}\; {{+ g}\; \delta_{j}}}}} \right)}} \right\}}}\end{matrix}$

So the same technique can be used with a simple redefinition of theconstants to be twice their original values. Similarly, the covariancebetween any two core assets can be calculated by simply addingcorresponding constants and repeating the same technique.

For the current parameter values, the constants for Bills, Bonds, andEquities are:

a b c d e F g Bills 0.0077 0 −1 0 1 0.7731 0 Bonds 0.0642 −2.5725−3.8523 0 2.5846 2.9031 0 Equities 0.0331 −2.4062 −3.7069 4.4431 2.482.79 −3.5487

Above, a methodology was described for calculating core asset analyticmoments for arbitrary horizons. This section describes how these momentsare translated into annualized moments. The procedure described in thissection essentially calculates successive annual moments for a twenty(20) year horizon and computes the arithmetic average of these moments.These ‘effective’ annual moments may then be used as inputs into thereverse optimization procedure and the individual optimization problem.

For this calculation, first make the following definitions:

M _(t) ^(j)=Expected return for j^(th) asset over the period t,t+1

Cov_(t) ^(i,j)=Covariance of returns on asset i with asset j over theperiod t,t+1

These expected returns and covariance are calculated using the formulasdescribed above The effective annual expected return for asset j is thencalculated as:

$M^{j} = {\sum\limits_{t = 1}^{T}\; {\omega_{t}M_{t}^{j}}}$

Similarly, the effective annual covariance between returns on asset iand returns on asset j are calculated as: (Note, the weights, ω_(t), arebetween zero and one, and sum to one.)

${Cov}^{i,j} = {\sum\limits_{t = 1}^{T}\; {\omega_{t}{Cov}_{t}^{i,j}}}$

In one embodiment, this annualizing technique could be personalized fora given user's situation. For example, the user's horizon could specifyT, and their level of current wealth and future contributions couldspecify the relevant weights. However for purposes of illustration, therelevant ‘effective’ moments for optimization and simulation arecomputed assuming a horizon of 20 years (T=20), and equal weights (i.e.1/T).

The techniques described in this section allow for the calculation ofthe following effective annual moments:

Output parameter name Description Units M¹ Bills: expected return Returnper year M² Bonds: expected return Return per year M³ Equity: expectedreturn Return per year Cov^(1,1) Bills: variance of returns (Return peryear)² Cov^(2,2) Bonds: variance of returns (Return per year)² Cov^(3,3)Equity: variance of returns (Return per year)² Cov^(1,2) Bills andBonds: covariance (Return per year)² Cov^(1,3) Bills and Equity:covariance (Return per year)² Cov^(2,3) Bonds and Equity: covariance(Return per year)²

Plan Monitoring

Exemplary conditions which may trigger an alert of some sort from theplan monitoring module 350 were described above. At this point, some ofthe real world events that may lead to those alert conditions will nowbe described. The real world events include the following: (1) afinancial product's style exposure changes, (2) the market value of theuser's assets have changed in a significant way, (3) new financialproducts become available to the user, (4) the risk characteristics ofthe user's portfolio have deviated from the desired risk exposure, or(5) the currently recommended portfolio no longer has the highestexpected return for the current level of portfolio risk (e.g., theportfolio is no longer on the mean-variance efficient frontier). Anefficient frontier is the sets of assets (portfolios) that provide thehighest level of return over different levels of risk. At each point onthe efficient frontier, there is no portfolio that provides a higherexpected return for the same or lower level of risk.

When a financial product's exposures change it may pull the user'sportfolio off of the efficient frontier. That is, due to a shift in theinvestment style of a particular financial product, the portfolio as awhole may no longer have the highest expected return for the currentlevel of risk. According to one embodiment of the present invention, ifthe inefficiency is greater than a predetermined tolerance or if theinefficiency will substantially impact one of the user's financialgoals, such as his/her retirement income goal, then the user is notifiedthat he/she should rebalance the portfolio. However, if the inefficiencyis within the predefined tolerance then the plan monitoring module 350may not alert the user. In one embodiment, the predefined tolerancedepends upon the impact of the inefficiency on expected wealth. Inaddition, the tolerance could depend upon relevant transaction costs.

A significant change in the market value of the user's assets may affectone or both of the probability of achieving a financial goal and thecurrent risk associated with the portfolio. In the case that the user'sportfolio has experienced a large loss, the portfolio may no longer bewithin a predetermined probability tolerance of achieving one or morefinancial goals. Further, as is typical in such situations, the riskassociated with the portfolio may also have changed significantly.Either of these conditions may cause the user to be notified thatchanges are required in the portfolio allocation or decision variablesto compensate for the reduction in market value of the portfolio. Alarge increase in the value of the user's portfolio, on the other hand,could trigger an alert due to the increase in the probability ofachieving one or more financial goals or due to the altered riskassociated with the newly inflated portfolio.

When one or more new financial products become available to the user,the user may be alerted by the plan monitoring module 350 if, forexample, a higher expected return may be possible at lower risk as aresult of diversifying the current portfolio to include one or more ofthe newly available financial products.

Having explained the potential effects of some real world events thatmay trigger alerts, exemplary plan monitoring processing will now bedescribed with respect to FIG. 8. At step 810, the data needed forreevaluating the current portfolio and for determining a current optimalportfolio is retrieved, such as the user profile and portfolio datawhich may be stored on the AdviceServer 110, for example. Importantly,the user profile may include investment plan profile information storedduring a previous session, such as the probability of reaching one ormore financial goals, the risk of the portfolio, and the like. Asdescribed above, selected user information on the AdviceServer 110 maybe kept up to date automatically if the financial advisory system 100has access to the record-keeping systems of the user's employer.Alternatively, selected user information may be updated manually by theuser.

At step 820, a current optimal portfolio is determined, as describedabove. Importantly, changes to the user database and/or portfolio dataare taken into consideration. For example, if one or more new financialproducts have become available to the user, portfolios including the oneor more new financial products are evaluated.

At step 830, the current portfolio is evaluated in a number of differentdimensions to determine if any trigger conditions are satisfied. Forexample, if the increase in expected wealth, or the increase in theprobability of reaching one or more investment goals resulting from areallocation to the current optimal portfolio is above a predeterminedtolerance, then processing will continue with step 840. Additionally, ifthe risk of the current portfolio is substantially different from theinvestment plan profile or if the probability of achieving one or morefinancial goals is substantially different from the investment planprofile, then processing continues with step 840.

At step 840, advice processing is performed. According to one embodimentof the present invention, based upon the user's preference among thedecision variables, the system may offer advice regarding which decisionvariable should be modified to bring the portfolio back on track toreach the one or more financial goals with the desired probability. Inaddition, the system may recommend a reallocation to improve efficiencyof the portfolio. An alert may be generated to notify the user of theadvice and/or need for affirmative action on his/her part. As describedabove, the alert may be displayed during a subsequent user session withthe financial advisory system 100 and/or the alerts may be transmittedimmediately to the user by telephone, fax, email, pager, fax, or similarmessaging system.

Advantageously, the plan monitoring module 350 performs ongoingportfolio evaluation to deal with the constantly changing data that mayultimately affect the exposure determination process and the portfoliooptimization process. In this manner, the user may receive timely adviceinstructing him/her how to most efficiently achieve one or morefinancial goals and/or maintain one or more portfolio characteristicsbased upon the available set of financial products.

In the foregoing specification, the invention has been described withreference to specific embodiments thereof. It will, however, be evidentthat various modifications and changes may be made thereto withoutdeparting from the broader spirit and scope of the invention. Thespecification and drawings are, accordingly, to be regarded in anillustrative rather than a restrictive sense.

1-24. (canceled)
 25. A method comprising: identifying a relationshipbetween returns of each financial product of a set of financial productsthat are available to a particular investor for investment and returnsof combinations of one or more factor asset classes of a set of factorasset classes by performing an exposure analysis on each financialproduct of the set of financial products; one or more processorsdetermining expected returns and volatility of returns for each of aplurality of efficient portfolios based upon the relationship, each ofthe plurality of efficient portfolios including a combination of one ormore of the financial products from the set of financial products; andidentifying a recommended portfolio of the plurality of efficientportfolios by selecting an efficient portfolio of the plurality ofefficient portfolios that maximizes an expected utility of wealth forthe particular investor.
 26. The method of claim 25, further comprising:forecasting returns associated with each core asset class of a set ofcore asset classes by generating core asset class scenarios based uponfuture scenarios of one or more economic factors with an equilibriumeconometric model; and forecasting returns associated with each factorasset class of the set of factor asset classes by generating factormodel asset scenarios based upon the core asset class scenarios.
 27. Themethod of claim 25, wherein said performing an exposure analysis on eachfinancial product of the set of financial products comprises performingreturns-based style analysis.
 28. The method of claim 25, wherein saidperforming an exposure analysis on each financial product of the set offinancial products comprises surveying the underlying assets held in thefinancial product.
 29. The method of claim 25, wherein said performingan exposure analysis on each financial product of the set of financialproducts comprises obtaining exposure information based on a targetbenchmark associated with the financial product.
 30. The method of claim25, wherein said performing an exposure analysis on each financialproduct of the set of financial products comprises categorizingexposures based on standard industry classification schemes.
 31. Themethod of claim 25, further comprising forecasting the probability ofthe particular investor meeting an identified financial goal based uponthe recommended efficient portfolio.
 32. The method of claim 25, furthercomprising receiving information indicative of the particular investor'srisk tolerance, and wherein said maximizing an expected utility ofwealth for the particular investor takes into consideration theinformation indicative of the particular investor's risk tolerance andinformation regarding expected returns and volatility of the expectedreturns for each of a plurality of efficient portfolios comprising acombination of one or more financial products from the set of financialproducts that is generated based upon the relationship.
 33. The methodof claim 25, wherein the set of financial products available to theparticular investor for investment comprises financial products offeredthrough an employee-directed defined contribution plan.
 34. Acomputer-readable storage medium containing a set of instructionscapable of causing one or more processors to: identify a relationshipbetween returns of each financial product of a set of financial productsthat are available to a particular investor for investment and returnsof combinations of one or more factor asset classes of a set of factorasset classes by performing an exposure analysis on each financialproduct of the set of financial products; determine expected returns andvolatility of returns for each of a plurality of efficient portfoliosbased upon the relationship, each of the plurality of efficientportfolios including a combination of one or more of the financialproducts from the set of financial products; and identify a recommendedportfolio of the plurality of efficient portfolios by selecting anefficient portfolio of the plurality of efficient portfolios thatmaximizes an expected utility of wealth for the particular investor.35-36. (canceled)
 37. A method comprising: a step for identifying arelationship between returns of each financial product of a set offinancial products that are available to a particular investor forinvestment and returns of combinations of one or more factor assetclasses of a set of factor asset classes; a step for determiningexpected returns and volatility of returns for each of a plurality ofefficient portfolios based upon the relationship, each of the pluralityof efficient portfolios including a combination of one or more of thefinancial products from the set of financial products; and a step foridentifying a recommended portfolio of the plurality of efficientportfolios.
 38. The method of claim 37, further comprising: a step forforecasting returns associated with each core asset class of a set ofcore asset classes by generating core asset class scenarios based uponfuture scenarios of one or more economic factors with an equilibriumeconometric model; and a step for forecasting returns associated witheach factor asset class of the set of factor asset classes based uponthe core asset class scenarios.
 39. A method comprising: forecastingreturns associated with each core asset class of a set of core assetclasses by generating core asset class scenarios based upon futurescenarios of one or more economic factors with an equilibriumeconometric model; one or more processors forecasting returns associatedwith each factor asset class of a set of factor asset classes bygenerating factor model asset scenarios based upon the core asset classscenarios; identifying a relationship between future returns of eachfinancial product of a set of financial products that are available to aparticular investor for investment and future returns of combinations ofone or more factor asset classes of the set of factor asset classes bydetermining each financial product's effective asset mix with respect tothe set of factor asset classes; determining expected returns andvolatility of returns for each of a plurality of efficient portfoliosbased upon the relationship, each of the plurality of efficientportfolios including a combination of one or more of the financialproducts from the set of financial products; and identifying arecommended portfolio of the plurality of efficient portfolios byselecting an efficient portfolio of the plurality of efficientportfolios that maximizes an expected utility of wealth for theparticular investor.
 40. A computer-readable storage medium containing aset of instructions capable of causing one or more processors to:forecast returns associated with each core asset class of a set of coreasset classes by generating core asset class scenarios based upon futurescenarios of one or more economic factors with an equilibriumeconometric model; forecast returns associated with each factor assetclass of a set of factor asset classes by generating factor model assetscenarios based upon the core asset class scenarios; identify arelationship between future returns of each financial product of a setof financial products that are available to a particular investor forinvestment and future returns of combinations of one or more factorasset classes of the set of factor asset classes by determining eachfinancial product's effective asset mix with respect to the set offactor asset classes; determine expected returns and volatility ofreturns for each of a plurality of efficient portfolios based upon therelationship, each of the plurality of efficient portfolios including acombination of one or more of the financial products from the set offinancial products; and identify a recommended portfolio of theplurality of efficient portfolios by selecting an efficient portfolio ofthe plurality of efficient portfolios that maximizes an expected utilityof wealth for the particular investor.